The Arithmetic Mean of the Divisors of an Integer
نویسندگان
چکیده
Dedicated to Emil Grosswald on the occasion of his sixty-eighth birthday .
منابع مشابه
A remark on the means of the number of divisors
We obtain the asymptotic expansion of the sequence with general term $frac{A_n}{G_n}$, where $A_n$ and $G_n$ are the arithmetic and geometric means of the numbers $d(1),d(2),dots,d(n)$, with $d(n)$ denoting the number of positive divisors of $n$. Also, we obtain some explicit bounds concerning $G_n$ and $frac{A_n}{G_n}$.
متن کاملIntegers without divisors from a fixed arithmetic progression
Let a be an integer and q a prime number. In this paper we find an asymptotic formula for the number of positive integers n ≤ x with the property that no divisor d > 1 of n lies in the arithmetic progression a modulo q. MSC numbers: 11A05, 11A25, 11N37
متن کاملOn certain arithmetic functions involving exponential divisors
The integer d is called an exponential divisor of n = ∏r i=1 p ai i > 1 if d = ∏r i=1 p ci i , where ci|ai for every 1 ≤ i ≤ r. The integers n = ∏r i=1 p ai i ,m = ∏r i=1 p bi i > 1 having the same prime factors are called exponentially coprime if (ai, bi) = 1 for every 1 ≤ i ≤ r. In this paper we investigate asymptotic properties of certain arithmetic functions involving exponential divisors a...
متن کاملOn the Coprimality of Some Arithmetic Functions
Let φ stand for the Euler function. Given a positive integer n, let σ(n) stand for the sum of the positive divisors of n and let τ(n) be the number of divisors of n. We obtain an asymptotic estimate for the counting function of the set {n : gcd(φ(n), τ(n)) = gcd(σ(n), τ(n)) = 1}. Moreover, setting l(n) := gcd(τ(n), τ(n+ 1)), we provide an asymptotic estimate for the size of #{n 6 x : l(n) = 1}.
متن کاملPrime divisors in Beatty sequences
We study the values of arithmetic functions taken on the elements of a non-homogeneous Beatty sequence αn+ β , n= 1,2, . . . , where α,β ∈R, and α > 0 is irrational. For example, we show that ∑ n N ω ( αn+ β )∼N log logN and ∑ n N (−1)Ω( αn+β ) = o(N), where Ω(k) and ω(k) denote the number of prime divisors of an integer k = 0 counted with and without multiplicities, respectively. © 2006 Elsevi...
متن کامل